Let us revisit the sequence tn of triangular numbers 1, 3, 6, 10, 15, … shown in the figure. Note that the nth term of this sequence is the sum of the first n natural numbers. Thus tn = n(n + 1)2. Can you use this to find the 10th, 17th and 80th triangular numbers?
Answer: t10 = 55, t17 = 153, t80 = 3240.
Step-by-step solution
Idea: The nth triangular number has rows of 1, 2, …, n dots, so it equals 1 + 2 + … + n = n(n + 1) ÷ 2. Substitute n = 10, 17, 80.
- t10 = 10 × 112 = 5 × 11 = 55.1 mark
- t17 = 17 × 182 = 17 × 9 = 153.1 mark
- t80 = 80 × 812 = 40 × 81 = 3240.1 mark
10th = 55, 17th = 153, 80th = 3240.
Check: t₁₀ = 1 + 2 + … + 10 = 55, which matches the pairing method ✓.
Answer to write in the exam
tn = n(n + 1)2
t10 = 10 × 112 = 55
t17 = 17 × 182 = 153
t80 = 80 × 812 = 3240
∴ 55, 153 and 3240
Common mistakes that cost marks
- Writing 17 × 18 ÷ 2 as 17 × 18 = 306 and forgetting to halve.
- Confusing triangular with square numbers and giving 102 = 100 for the 10th.
- Halving both factors (e.g. 40 × 40.5). Halve only one of them.
How this can come in the exam
MCQ (1 mark)
The 24th triangular number is
- 276
- 300
- 288
- 552
Show answer
(B) 300
24 × 25 ÷ 2 = 300.
Try one yourself
Find the 15th and the 90th triangular numbers.
Show answer
t15 = 15 × 16 ÷ 2 = 120; t90 = 90 × 91 ÷ 2 = 4095.
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